Minimization of Risk and Linear Quadratic Optimal Control Theory

Abstract

This article is concerned with the optimal control problem for the linear stochastic system $X_t=x+\int_0^t(A_sX_s+B_su_s+f_s)\, ds+\int_0^t\sum_{i=1}^d[C_i(s)X_s+D_i(s)u_s+g_i(s)]\, dw_i(s)$ with the convex risk functional $J(u)=EM(X_T)+E\int_0^TG(t, X_t,u_t)\, dt$. In order to guarantee the existence of an optimal control without any (weak) compactness assumption on the admissible control set, we assume that the risk function M is coercive and that $\sum_{i=1}^dD_i^*D_i$ is uniformly positive, rather than to assume like in the control literature that the running risk function G is coercive with respect to the control variable. In this new setting, the running risk function G may be independent of the control variable, and therefore the so-called singular linear-quadratic (LQ) stochastic control problem is included. A rigorous theory is developed for the general stochastic LQ problem with random coefficients, and the bounded mean oscillation--martingale theory is used to account for the concerned integrability. It plays a crucial role in the following exposition: (a) to connect the stochastic LQ problem to two associated backward stochastic differential equations (BSDEs)---one is an n × n symmetric matrix-valued nonlinear Riccati BSDE and the other is an n-dimensional linear BSDE with unbounded coefficients; (b) to show that the latter BSDE has an adapted solution pair of the suitably necessary regularity. This seems to be the first application in a stochastic LQ theory of the BMO-martingale theory, which roots in harmonic analysis. Furthermore, with the help of an a priori estimate on the risk functional, existence and uniqueness of the solutions of backward stochastic Riccati differential equations (BSRDEs) in the singular case is reduced to the regular case via a perturbation method, and then a new existence and uniqueness result on BSRDEs is obtained for the singular case.